Almansi Decomposition for Dunkl-Helmholtz Operators

Abstract

We consider the iterated Dunkl-Helmholtz equation (Δ_h − γ)ⁿ f = 0 for nonzero γ in a domain of ℝ^N. Here Δ_h = Σ ^N _j=1 D ²_j is the Dunkl Laplacian, and D _j is the Dunkl operator attached to the Coxeter group G associated with the reduced root system R,

$$ \mathcal{D}_j f(x) = \frac{{\partial f}} {{\partial x_j }}(x) + \sum\limits_{v \in R_ + } {k_v \frac{{f(x) - f(\sigma _v x)}} {{\langle x,v\rangle }}} v_j , $$

where κ _v is a multiplicity function on R and σ _v is the reflection with respect to the root v.

We prove that any solution f of the iterated Dunkl-Helmholtz equation has a decomposition of the form

$$ f(x) = f_0 (x) + R_\mu f_1 (x) + \cdots + R_\mu ^{n - 1} f_{n - 1} (x),\forall x \in \Omega , $$

where f _j are annihilated by Δ_h − γ, μ is a fixed but arbitrary complex number, and R ⁿ_μ = (R _μ)ⁿ are given by R _μ = μI + R ₀, with I the identity operator and R ₀ the Euler operator.

The first author was partially supported by the NNSF of China (No. 10471134), the SRFDP 20050358052, and NCET. The second author was partially supported by the R&D unit Matemática a Aplicações (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), co-financed by the European Community fund FEDER.